III-85 



where the surface S is generated by lines parallel to the y axis and 





is the maximum value of the slope. The total field p = p. + p at an arbitrary 



point (x, z) can be expressed in terms of the field and its normal derivative on the 

 scattering surface, using Green's formula. For the two dimensional case, this 

 is given by 



'II 



p(x, z) = p^^^(x, z) + j J ; p ^ H^<« (kr,) - H^<^>(lcr,)|E ds (111-98) 



s 



where r-y is the distance between the point of observation (x, z) and the variable 

 point (xi , S(xi ) ) on the surface . The integration is along the curve obtained by 

 cutting the uneven surface S(x) by the plane y = . 



The boundary condition used in this model is the same as in the others, 

 i.e., p = for z = S(x) . Then, the scattered wave is given by 



4 1 o 



Pgc = " i \ ^-^^^ ^^^1^ f(xi)dxi (III-99a) 



where 



f(xO = ^ i- or ^ 4^ (III-99b) 



dn n n dn 



z z 



For this we must know f(xi) . This is obtained by solving 



k\< 



p.^^(x3, S(X3)) - ^ I H^^^^ (k ri)f(xi) dxi (III-lOO) 



where (xg, S(x2)) is another arbitrary point on the surface, and ri is the distance 

 between (xi, S(xi)) and (xg, S(x2) ). 



Using the approximation (III-97), (III-lOO) can be written as: 



p.^^(x2, S(X3)) =i- H^^^^ (k|x3-xi| ) f(xi)dxi (III-lOl) 



In order to solve this expression for f(xi) explicitly, suppose the incident wave is a 

 plane wave of the form 



ik(ax+Yz) (III- 102) 



p. (x, z) = e 

 mc 



Arthur H.littlfJnir. 



S-7001-0307 



